Fix a finite interval $I$.It suffices to show that almost surely,
$$\mathcal L(\{t \in I \, | \, B_t = A_t \}) = 0 \,.$$
Brownian motion restricted to $\{t \in I \, | \, B_t = A_t \}$ has bounded variation, so a positive answer is implied by the following stronger result, a special case of Theorem 1.3 of [1]:

**Theorem[1]** Let $\{B(t): t\in [0,1]\}$ be a standard Brownian motion.
Then, almost surely, for all $S\subset [0,1]$,
if $B|_{S}$ is of bounded variation, then $\overline{\dim}_M S\leq 1/2$.

Here $\overline{\dim}_M$ is the upper Minkowski dimension (a.k.a. upper box dimension.)

[1] Angel, Omer, Richárd Balka, András Máthé, and Yuval Peres. "Restrictions of Hölder continuous functions." Transactions of the American Mathematical Society 370, no. 6 (2018): 4223-4247.
https://www.ams.org/journals/tran/2018-370-06/S0002-9947-2018-07126-4/S0002-9947-2018-07126-4.pdf
http://wrap.warwick.ac.uk/84253/7/WRAP-restrictions-continuous-functions-Peres-2018.pdf